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One-Dimensional Site Response Analysis What do we mean? One-dimensional = Waves propagate in one direction only

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One-Dimensional Site Response Analysis What do we mean? One-dimensional = waves propagate in one direction only Motion is identical on planes perpendicular to that motion to infinity

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One-Dimensional Site Response Analysis What do we mean? One-dimensional = waves propagate in one direction only Motion is identical on planes perpendicular to that motion Cant handle refraction so layer boundaries must be perpendicular to direction of wave propagation Usual assumption is vertically-propagating shear (SH) waves Horizontal input motion Horizontal surface motion

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One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Stiffer with depth Focus

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One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Stiffer with depth Horizontal boundaries – waves tend to be refracted toward vertical Decreasing stiffness causes refraction of waves to increasingly vertical path Focus

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One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Stiffer with depth Not appropriate here

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Retaining structures Dams and embankments Dams and embankments Tunnels One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Inclined ground surface and/or non- horizontal boundaries can require use of two-dimensional analyses Not here!

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Complex soil conditions Complex soil conditions Dams in narrow canyons Dams in narrow canyons Multiple structures Multiple structures One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Localized structures may require use of 3-D response analyses Not here!

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One-Dimensional Site Response Analysis How should ground motions be applied? Incoming motion uiui Rock outcropping motion 2u i Bedrock motion u i + u r Free surface motion usus Not the same! Soil Rock

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One-Dimensional Site Response Analysis How should ground motions be applied? Object motion Free surface motion usus Input (object) motion If recorded at rock outcrop, apply as outcrop motion (program will remove free surface effect). Bedrock should be modeled as an elastic half-space. If recorded in boring, apply as within- profile motion (recording does not include free surface effect). Bedrock should be modeled as rigid.

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Complex Response Method Approach used in computer programs like SHAKE Transfer function is used with input motion to compute surface motion (convolution) For layered profiles, transfer function is built layer-by-layer to go from input motion to surface motion Amplification De-amplification Methods of One-Dimensional Site Response Analysis Single elastic layer

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Layer j+1 Layer j Consider the soil deposit shown to the right. Within a given layer, say Layer j, the horizontal displacements will be given by At the boundary between layer j and layer j+1, compatibility of displacements requires that Continuity of shear stresses requires that Complex Response Method (Linear analysis) Amplitudes of upward- and downward-traveling waves in Layer j Equilibrium satisfied No slip

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Defining * j as the complex impedance ratio at the boundary between layers j and j+1, the wave amplitudes for layer j+1 can be obtained from the amplitudes of layer j by solving the previous two equations simultaneously Wave amplitudes in Layer j Wave amplitudes in Layer j+1 So, if we can go from Layer j to Layer j+1, we can go from j+1 to j+2, etc. This means we can apply this relationship recursively and express the amplitudes in any layer as functions of the amplitudes in any other layer. We can therefore build a transfer function by repeated application of the above equations. Complex Response Method (Linear analysis) Propagation of wave energy from one layer to another is controlled by (complex) impedance ratio

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Complex Response Method (Linear analysis) Single layer on rigid base H = 100 ft V s = 500 ft/sec = 10% Single layer on rigid base H = 100 ft V s = 500 ft/sec = 10%

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Complex Response Method (Linear analysis) Single layer on rigid base H = 50 ft V s = 1,500 ft/sec = 10% Single layer on rigid base H = 50 ft V s = 1,500 ft/sec = 10%

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Complex Response Method (Linear analysis) Single layer on rigid base H = 100 ft V s = 300 ft/sec = 5% Single layer on rigid base H = 100 ft V s = 300 ft/sec = 5%

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Complex Response Method (Linear analysis)

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Different sequence of soil layers Different transfer function Different response

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Complex Response Method (Linear analysis) Another sequence of soil layers Different transfer function Different response

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Complex Response Method (Linear analysis) Complex response method operates in frequency domain Input motion represented as sum of series of sine waves Solution for each sine wave obtained Solutions added together to get total response Principle of superposition Linear system Can we capture important effects of nonlinearity with linear model?

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Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent shear modulus Equivalent damping ratio Equivalent Linear Approach

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Assume some initial strain and use to estimate G and (1) Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach

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(1) Use these values to compute response Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach (t) t

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(1) Determine peak strain and effective strain eff = R max Determine peak strain and effective strain eff = R max Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach (t) t max eff

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(1) (2) Select properties based on updated strain level Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach

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(1) (2) (3) Compute response with new properties and determine resulting effective shear strain Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach

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Repeat until computed effective strains are consistent with assumed effective strains eff Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. Equivalent Linear Approach

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Advantages: Can work in frequency domain Compute transfer function at relatively small number of frequencies (compared to doing calculations at all time steps) Increased speed not that significant for 1-D analyses Increased speed can be significant for 2-D, 3-D analyses Equivalent linear properties readily available for many soils – familiarity breeds comfort/confidence Can make first-order approximation to effects of nonlinearity and inelasticity within framework of a linear model Equivalent Linear Approach The equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately. … often, a very good one!

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z Divide profile into series of layers Divide time into series of time steps t

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z Divide profile into series of layers Divide time into series of time steps t v ij = v (z = z i, t = t j ) tjtj zizi

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response.

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response.

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response.

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response.

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Nonlinear Analysis Equation of motion must be integrated in time domain Wave equation for visco-elastic medium z t tjtj zizi More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. Procedure steps through time from beginning of earthquake to end. Step through time

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Nonlinear Behavior Continuous Linear segments ActualApproximation In a nonlinear analysis, we approximate the continuous actual stress-strain behavior with an incrementally-linear model. The finer our computational interval, the better the approximation.

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Advantages: Work in time domain Can change properties after each time step to model nonlinearity Can formulate model in terms of effective stresses Can compute pore pressure generation Can compute pore pressure redistribution, dissipation Avoids spurious resonances (associated with linearity of EL approach) Can compute permanent strain permanent deformations Nonlinear Approach Liquefaction Nonlinear analyses can produce results that are consistent with equivalent linear analyses when strains are small to moderate, and more accurate results when strains are large. They can also do important things that equivalent linear analyses cant, such as compute pore pressures and permanent deformations. Nonlinear analyses can produce results that are consistent with equivalent linear analyses when strains are small to moderate, and more accurate results when strains are large. They can also do important things that equivalent linear analyses cant, such as compute pore pressures and permanent deformations.

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What are people using in practice? Equivalent Linear vs. Nonlinear Approaches Equivalent linear analyses One-dimensional – 2-D / 3-D – Nonlinear analyses One-dimensional – 2-D / 3-D – SHAKE QUAD4, FLUSH DESRA, DMOD TARA, FLAC, PLAXIS

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What are people using in practice? Equivalent Linear vs. Nonlinear Approaches Equivalent linear analyses One-dimensional – 2-D / 3-D – Nonlinear analyses One-dimensional – 2-D / 3-D – SHAKE QUAD4, FLUSH DESRA TARA

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DimensionsOSEquivalent LinearNonlinear 1-D DOSDyneq, Shake91AMPLE, DESRA, DMOD, FLIP, SUMDES, TESS WindowsShakeEdit, ProShake, Shake2000, EERA CyberQuake, DeepSoil, NERA, FLAC, DMOD2000 2-D / 3-D DOS FLUSH, QUAD4/QUAD4M, TLUSH DYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSees WindowsQUAKE/W, SASSI2000FLAC, PLAXIS Available Codes Since early 1970s, numerous computer programs developed for site response analysis Can be categorized according to computational procedure, number of dimensions, and operating system

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Current Practice Informal survey developed to obtain input on site response modeling approaches actually used in practice Emailed to 204 people Attendees at ICSDEE/ICEGE Berkeley conference (non-academic) Geotechnical EERI members – 2003 Roster (non-academic) Survey Respondents WNAENAOverseas PrivatePublicPrivatePublicPrivatePublic Number of responses3536155 55 responses Western North America (WNA) Eastern North America (ENA) Overseas Private firms Public agencies

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Current Practice Method of Analysis Method of Analysis WNAENAOverseas Private (35) Public (3) Private (6) Public (1) Private (5) Public (5) 1-D Equivalent Linear68528650245 1-D Nonlinear1117120485 2-D/3-D Equiv. Linear92812560 2-D/3-D Nonlinear1231252390 Of the total number of site response analyses you perform, indicate the approximate percentages that fall within each of the following categories: [ ] a. One-dimensional equivalent linear [ ] b. One-dimensional nonlinear [ ] c. Two- or three-dimensional equivalent linear [ ] d. Two- or three-dimensional nonlinear One-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? 30 m u(H,t)u(H,t) u(0,t) V s = 300 m/sec V s = 762 m/sec 1 m 15 m 29 m Topanga record (Northridge) T s = 0.4 sec

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response

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Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Relatively low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Relatively low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration Velocity

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Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Relatively low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Equivalent linear overpredicts nonlinear response at certain frequencies – spurious resonances Stress-strain response becoming more complicated – more variable stiffness and less elliptical shape

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Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Relatively low degree of nonlinearity Similar response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Stiffness starting to vary more significantly over course of ground motion

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Topanga motion scaled to 0.50 g Strong motion + stiff soil Moderate strains Low – moderate degree of nonlinearity Noticeably different response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration

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Topanga motion scaled to 0.50 g Strong motion + stiff soil Moderate strains Low – moderate degree of nonlinearity Noticeably different response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Topanga motion scaled to 1.0 g Very strong motion + stiff soil Moderate strains Moderate degree of nonlinearity Noticeably different response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration Substantial softening by EL method causes underprediction of initial portion of record Linearity inherent in EL method causes overprediction response in strongest portion of record Softening by EL method causes underprediction

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Topanga motion scaled to 0.50 g Very strong motion + stiff soil Moderate strains Moderate degree of nonlinearity Noticeably different response Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make?

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? 14 m V s = 300 m/sec V s = 762 m/sec 16 m V s = 100 m/sec u(H,t)u(H,t) u(0,t) 1 m 15 m 29 m

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles Acceleration EL model predicts very soft behavior at beginning of earthquake, before any large strains have developed.

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles Acceleration More consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles – too much?

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles Acceleration NL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout.

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Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles

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Time Nonlinear Soil Behavior Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002) Low stiffness High stiffness Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations arent seen.

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Time Nonlinear Soil Behavior Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002) High damping Low damping Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations arent seen.

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High frequencies are associated with smaller strains High stiffness and low damping are associated with smaller strains Make stiffness and damping frequency-dependent Modified Equivalent Linear Approach Normalized strain spectra from five motions Normalized strain spectrum from one motion Three orders of magnitude Frequency (Hz)

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Assimaki and Kausel Modified Equivalent Linear Approach Frequency-dependent model Conventional model High frequencies oversoftened and overdamped Excellent agreement with nonlinear model

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Benchmarking of Nonlinear Analyses Stewart and Kwok PEER study to determine proper manner in which to use nonlinear analyses Worked with five existing nonlinear codes; hired developers to run their codes and comment on results Established advisory committee to oversee analyses and assist with interpretation Met regularly with advisory committee and developers

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Benchmarking of Nonlinear Analyses Stewart and Kwok Considered codes

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D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Damping ratio Frequency Mass-proportional Stiffness-proportional Rayleigh Benchmarking of Nonlinear Analyses

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D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark method for time integration Variable slice width – simulating response of dams, embankments on rock Benchmarking of Nonlinear Analyses Decreasing stiffness due to geometry

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D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark method for time integration Variable slice width – simulating response of dams, embankments on rock Can simulate slip on weak interfaces Uses MKZ soil model (modified hyperbola – needs G max, max, and s) Can soften backbone curve to model cyclic degradation Benchmarking of Nonlinear Analyses

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D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark method for time integration Variable slice width – simulating response of dams, embankments on rock Can simulate slip on weak interfaces Uses MKZ soil model (modified hyperbola – needs G max, max, and s) Can soften backbone curve to model cyclic degradation Uses Masing rules for unloading-reloading behavior Need input parameters for: MKZ backbone curve (4) Cyclic degradation (3 for clay, 4 for sand) Pore pressure generation (4 for clay, 4 for sand) Pore pressure redistribution/dissipation (at least 2) Rayleigh damping coefficients (2) Basic layer properties (density, shear wave velocity, half-space properties) Need input parameters for: MKZ backbone curve (4) Cyclic degradation (3 for clay, 4 for sand) Pore pressure generation (4 for clay, 4 for sand) Pore pressure redistribution/dissipation (at least 2) Rayleigh damping coefficients (2) Basic layer properties (density, shear wave velocity, half-space properties) Benchmarking of Nonlinear Analyses

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DEEPSOIL (Hashash) Similar to DMOD-2 (lumped mass, derives from DESRA-2) More advanced Rayleigh damping scheme (lower frequency dependence) TESS (Pyke) Finite difference wave propagation analysis (not lumped mass) Cundall-Pyke hypothesis for loading-unloading behavior Similar backbone curve to DMOD-2 and DEEPSOIL Inviscid (sort of) low-strain damping scheme OpenSees (Yang, Elgamal) Finite element model (1D, 2D, 3D capabilities) Multi-surface plasticity model (von Mises yield surface, kinematic hardening, non-associative flow rule) Full Rayleigh damping SUMDES Finite element model Bounding surface plasticity model (Lade-like yield surface, kinematic hardening, non-associative flow rule) Simplified Rayleigh damping Benchmarking of Nonlinear Analyses

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Recommendations Specification of control motion For outcropping motion, use recorded motion with elastic base For motions recorded at depth, use recorded motion with rigid base Specification of viscous damping Use full or extended Rayleigh damping – iterate on selection of control frequencies to match equivalent linear response for low loading levels (linear response domain). If not possible, use full Rayleigh damping with targets at f o and 5f o. Backbone curve parameters Adjust, if possible, to produce correct shear strength at large strains Bound nonlinear, inelastic behavior by running analyses with: Backbone curve fit to match G/G max behavior Backbone curve fit to minimize error in G/G max and damping curves Benchmarking of Nonlinear Analyses

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Performance Based on validations against vertical array data Models produce reasonable results Some indication of overdamping at high frequencies, overamplification at site frequency Variability of predictions due to backbone curves and damping models most pronounced at T<0.5 sec and is significant only for relatively thick profiles. Model-to-model variability most pronounced at low periods. Nonlinearity modeled well up to levels for which adequate data is available (generally up to about 0.2g). Data for stronger shaking being sought (centrifuge tests, recent Nigaata earthquake). DMOD-2, DEEPSOIL, and OpenSees generally produced similar amplification factors and spectral shapes; TESS produced different response at high frequencies (different damping formulation), SUMDES results were significantly different than all others for deep sites (probably due to simplified Rayleigh damping). Benchmarking of Nonlinear Analyses

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Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

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Nonlinear Behavior– Effective Stress Analyses Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

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Nonlinear Behavior– Effective Stress Analyses Nonlinear Behavior – Effective Stress Analyses Wildlife – Elmore Ranch recordings

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Nonlinear Behavior– Effective Stress Analyses Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings Low frequency High frequency Ground surface record ???

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Site Effects Elmore Ranch record – no liquefaction Ratio of wavelet amplitudes – variation with frequency and time Time (sec) Frequency (Hz)

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Site Effects Elmore Ranch record – no liquefaction Ratio of wavelet amplitudes – variation with frequency and time Time (sec) Frequency (Hz)

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Nonlinear Behavior– Effective Stress Analyses Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

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Nonlinear Behavior– Effective Stress Analyses Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

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One-Dimensional Site Response Analysis Summary Must be aware of assumptions Uni-directional wave propagation (normal to layer boundaries) Uni-directional particle motion (no surface waves) Particularly useful for profiles with high impedance contrasts Equivalent linear approach works very well for most cases Material properties readily available Computations performed rapidly Nonlinear analyses match equivalent linear when strains are small Nonlinear analyses are preferred when strains are high – soft soils and/or strong shaking Can account for shear strength of soil Can handle pore pressure generation – some well, some poorly Can predict permanent deformations – for common for 2-D analyses

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